Synchronization of Dirac-Bianconi driven oscillators

TL;DR

The paper addresses synchronization in higher-order networks where dynamics reside on both nodes and edges and interact via the Dirac-Bianconi operator $\mathcal{D}$. It defines Dirac-Bianconi driven oscillators using topological signals on consecutive dimensions and analyzes two coupled oscillators through phase reduction, revealing that coupling through link signals markedly enhances weak-coupling synchronization. A FitzHugh-Nagumo-inspired DBFHN realization demonstrates a robust limit cycle driven by $\mathcal{D}$, and phase-sensitivity analysis shows $1$-cochains (edges) dominate the phase response, aligning with the observed ease of synchronization under Dirac-Bianconi coupling. The work provides a tractable framework for higher-order dynamics with potential applications to brain-edge signaling and sets the stage for extensions to higher dimensions, directed simplicial structures, stochastic dynamics, and control strategies.

Abstract

In dynamical systems on networks, one assigns the dynamics to nodes, which are then coupled via links. This approach does not account for group interactions and dynamics on links and other higher dimensional structures. Higher-order network theory addresses this by considering variables defined on nodes, links, triangles, and higher-order simplices, called topological signals (or cochains). Moreover, topological signals of different dimensions can interact through the Dirac-Bianconi operator, which allows coupling between topological signals defined, for example, on nodes and links. Such interactions can induce various dynamical behaviors, for example, periodic oscillations. The oscillating system consists of topological signals on nodes and links whose dynamics are driven by the Dirac-Bianconi coupling, hence, which we call it Dirac-Bianconi driven oscillator. Using the phase reduction method, we obtain a phase description of this system and apply it to the study of synchronization between two such oscillators. This approach offers a way to analyze oscillatory behaviors in higher-order networks beyond the node-based paradigm, while providing a ductile modeling tool for node- and edge-signals.

Figures (8)

• Figure 1: Visualization of a Dirac-Bianconi driven oscillator: the $0$-cochains $u_i$ and $1$-cochains $v_i$, grouped in the topological spinor $\vec{w}$, are isolated and do not feel each other's presence until the Dirac-Bianconi operator $\mathcal{D}$ couples them together, giving rise to a network whose dynamics is periodic. The oscillations are the results or the interactions between adjacent $0$- and $1$-cochains through the boundary and co-boundary operators $\boldsymbol{B}_1$ and $\boldsymbol{B}_1^\top$, i.e., the Dirac-Bianconi operator.
• Figure 2: Dynamics of the Dirac-Bianconi driven oscillator described in Eq. \ref{['eq:DBFHN']}: in panel a), we depict the time series of the $0$- and $1$-cochains; after a transient, the system follows a periodic trajectory of period $T=21.14$ time units (t.u.). Panel b) shows a snapshot of the values of the signals on nodes and links for $t=290$ t.u. The parameters are the following: $a_1=a_2=0.7$, $a_3=-3$, $b_1=b_2=b_3=0.3$, $\delta_1=\delta_2=\delta_3=0.08$, $I_1=I_2=I_3=0.8$. Note that $b_j$, $\delta_j$ and $I_i$ do not have to be equal for every system to have a periodic solution.
• Figure 3: Dynamics of each unit of the Dirac-Bianconi driven oscillator over one period:$\boldsymbol{B}$ is such that the system has a homogeneous state and the model parameters are $a_1=a_3=0.7$, $a_2=-3$, $b_1=b_2=b_3=0.3$, $\delta_1=\delta_2=\delta_3=0.08$, $I_1=I_2=I_3=0.8$ (same setting of the previous Figure). The resulting period of the Dirac-Bianconi driven oscillator is $T=21.14$ t.u.
• Figure 4: Weak interaction between two Dirac-Bianconi driven oscillators: pictorial representation of the setting considered in the text, namely, two Dirac-Bianconi driven oscillators weakly coupled via a pair of nodes. The pairwise coupling is, moreover, considered as linear and diffusive-like. Of course, more complex coupling configurations can be considered. Although it may seem that no higher-order effects take place in the coupling, synchronization between the two oscillators can be achieved only when considering the effects of the $1$-cochains, i.e., the variables on the links.
• Figure 5: Synchronization of two Dirac-Bianconi driven oscillators with weak diffusive coupling: we show the frequencies of DB1 (blue) and DB2 (red), coupled as in Eqs. \ref{['eq:DBFHN1_incomplete']} and \ref{['eq:DBFHN2_incomplete']} integrated until $t=6000$ t.u. and averaged over $100$ periods. We can observe that, with diffusive coupling, synchronization is achieved only for large values of the coupling strength. The parameters are $\bar{a}_1=\bar{a}_3=\tilde{a}_1=\tilde{a}_3=0.7$, $\bar{a}_2=\tilde{a}_2=-3$, $\bar{b}_1=\bar{b}_2=\bar{b}_3=\tilde{b}_1=\tilde{b}_2=\tilde{b}_3=0.3$, $\bar{\delta}_1=\bar{\delta}_2=\bar{\delta}_3=\tilde{\delta}_1=\tilde{\delta}_2=\tilde{\delta}_3=0.08$, $\bar{I}_1=\bar{I}_2=\bar{I}_3=\tilde{I}_1=\tilde{I}_2=0.8$ and $\tilde{I}_3=0.4$.